Download Lesson 7

Properties of Addition
Property
Additive Identity
Element
Definition
The sum of a number and
zero is that number.
Commutative Property
Associative Property
Additive Inverses
(opposites)
The sum of a number and its
opposite is 0.
Closure Property
The sum of any two numbers
is a unique real number.
Example
Properties of Multiplication
Property
Multiplicative Identity
Element
Definition
The product of a number and
one is that number.
Commutative Property
Associative Property
Multiplicative Inverses
(Reciprocals)
When the product of two
numbers is one.
Multiplication Property
of Zero
Closure Property
The product of any two numbers
is a unique real number
Example
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
Lesson 7A: Algebraic Expressions—The Commutative and
Associative Properties
Classwork
Four Properties of Arithmetic:
The Commutative Property of Addition: If 𝑎 and 𝑏 are real numbers, then 𝑎 + 𝑏 = 𝑏 + 𝑎.
The Associative Property of Addition: If 𝑎, 𝑏, and 𝑐 are real numbers, then (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐)
The Commutative Property of Multiplication: If 𝑎 and 𝑏 are real numbers, then 𝑎 × 𝑏 = 𝑏 × 𝑎.
The Associative Property of Multiplication: If 𝑎, 𝑏, and 𝑐 are real numbers, then (𝑎𝑏)𝑐 = 𝑎(𝑏𝑐).
Exercise 1
Viewing the diagram below from two different perspectives illustrates that (3 + 4) + 2 equals
2 + (4 + 3).
Is it true for all real numbers 𝑥, 𝑦 and 𝑧 that (𝑥 + 𝑦) + 𝑧 should equal (𝑧 + 𝑦) + 𝑥?
(Note: The direct application of the Associative Property of Addition only gives(𝑥 + 𝑦) + 𝑧 = 𝑥 + (𝑦 + 𝑧).)
Exercise 2
Draw a flow diagram and use it to prove that (𝑥𝑦)𝑧 = (𝑧𝑦)𝑥 for all real numbers 𝑥, 𝑦, and 𝑧.
Lesson 7:
Date:
Algebraic Expressions—The Commutative and Associative Properties
4/30/17
S.31
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
31
Lesson #:
Lesson Description
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
Exercise 3
Use these abbreviations for the properties of real numbers and complete the flow diagram.
𝐶+ for the commutative property of addition
𝐶× for the commutative property of multiplication
𝐴+ for the associative property of addition
𝐴× for the associative property of multiplication
Exercise 4
Let 𝑎, 𝑏, 𝑐, and 𝑑 be real numbers. Fill in the missing term of the following diagram to show that ((𝑎 + 𝑏) + 𝑐) + 𝑑 is
sure to equal 𝑎 + (𝑏 + (𝑐 + 𝑑)).
Lesson 7:
Date:
Algebraic Expressions—The Commutative and Associative Properties
4/30/17
S.32
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
32
Lesson #:
Lesson Description
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
Important Definitions
NUMERICAL SYMBOL: A numerical symbol is a symbol that represents a specific number.
For example, 0, 1, 2, 3,
2
, −3, −124.122, 𝜋, 𝑒 are numerical symbols used to represent specific points on the real number
3
line.
VARIABLE SYMBOL: A variable symbol is a symbol that is a placeholder for a number.
It is possible that a question may restrict the type of number that a placeholder might permit, e.g., integers only or
positive real number.
ALGEBRAIC EXPRESSION: An algebraic expression is either
1.
a numerical symbol or a variable symbol, or
2.
the result of placing previously generated algebraic expressions into the two blanks of one of the four operators
((__) + (__), (__) − (__), (__) × (__), (__) ÷ (__)) or into the base blank of an exponentiation with exponent that is
a rational number.
Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the
Commutative, Associative, and Distributive Properties and the properties of rational exponents to components of the
first expression.
NUMERICAL EXPRESSION: A numerical expression is an algebraic expression that contains only numerical symbols (no
variable symbols), which evaluate to a single number.
The expression, 3 ÷ 0, is not a numerical expression.
EQUIVALENT NUMERICAL EXPRESSIONS: Two numerical expressions are equivalent if they evaluate to the same
number.
Note that 1 + 2 + 3 and 1 × 2 × 3, for example, are equivalent numerical expressions (they are both 6) but 𝑎 + 𝑏 + 𝑐
and 𝑎 × 𝑏 × 𝑐 are not equivalent expressions.
Lesson Summary
The Commutative and Associative Properties represent key beliefs about the arithmetic of real numbers. These
properties can be applied to algebraic expressions using variables that represent real numbers.
Two algebraic expressions are equivalent if we can convert one expression into the other by repeatedly applying the
Commutative, Associative, and Distributive Properties and the properties of rational exponents to components of
the first expression.
Lesson 7:
Date:
Algebraic Expressions—The Commutative and Associative Properties
4/30/17
S.33
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
33
Lesson #:
Lesson Description
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
Problem Set
1.
The following portion of a flow diagram shows that the expression 𝒂𝒃 + 𝒄𝒅 is equivalent to the expression 𝒅𝒄 + 𝒃𝒂.
Fill in each circle with the appropriate symbol: Either 𝐶+ (for the “Commutative Property of Addition”) or 𝐶× (for the
“Commutative Property of Multiplication”).
2.
Fill in the blanks of this proof showing that (𝑤 + 5)(𝑤 + 2) is equivalent 𝑤 2 + 7𝑤 + 10. Write either
“Commutative Property,” “Associative Property,” or “Distributive Property” in each blank.
(𝑤 + 5)(𝑤 + 2)
= (𝑤 + 5)𝑤 + (𝑤 + 5) × 2
= 𝑤(𝑤 + 5) + (𝑤 + 5) × 2
= 𝑤(𝑤 + 5) + 2(𝑤 + 5)
= 𝑤 2 + 𝑤 × 5 + 2(𝑤 + 5)
= 𝑤 2 + 5𝑤 + 2(𝑤 + 5)
= 𝑤 2 + 5𝑤 + 2𝑤 + 10
= 𝑤 2 + (5𝑤 + 2𝑤) + 10
= 𝑤 2 + 7𝑤 + 10
3.
____________________________
What is a quick way to see that the value of the sum 53 + 18 + 47 + 82 is 200?
Lesson 7:
Date:
Algebraic Expressions—The Commutative and Associative Properties
4/30/17
S.34
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
34
Lesson #:
Lesson Description
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
4.
Fill in each circle of the following flow diagram with one of the letters: C for Commutative Property (for either
addition or multiplication), A for Associative Property (for either addition or multiplication), or D for Distributive
Property.
5.
The following is a proof of the algebraic equivalency of (2𝑥)3 and 8𝑥 3 . Fill in each of the blanks with either the
statement “Commutative Property” or “Associative Property.”
(2𝑥)3 = 2𝑥 ∙ 2𝑥 ∙ 2x
Power to a Power___________
= 2(𝑥 × 2)(𝑥 × 2)𝑥
_________________________
= 2(2𝑥)(2𝑥)𝑥
_________________________
= 2 ∙ 2(𝑥 × 2)𝑥 ∙ 𝑥
_________________________
= 2 ∙ 2(2𝑥)𝑥 ∙ 𝑥
_________________________
= (2 ∙ 2 ∙ 2)(𝑥 ∙ 𝑥 ∙ 𝑥)
_________________________
= 8𝑥 3
Lesson 7:
Date:
Algebraic Expressions—The Commutative and Associative Properties
4/30/17
S.35
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
35
Lesson #:
Lesson Description
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
6.
Write a mathematical proof of the algebraic equivalency of (𝑎𝑏)2 and 𝑎2 𝑏 2 . (I provided a start for you there are
three more steps to the conclusion)
(𝒂𝒃)𝟐 = (𝒂𝒃)(𝒂𝒃)
7.
________________________________
a. Suppose we are to play the 4-number game with the symbols a, b, c, and d to represent numbers, each used at
most once, combined by the operation of addition ONLY. If we acknowledge that parentheses are not needed, show
there are essentially only 15 expressions one can write.
b. How many answers are there for the multiplication ONLY version of this game?
8.
Write a mathematical proof to show that (𝑥 + 𝑎)(𝑥 + 𝑏) is equivalent to 𝑥 2 + 𝑎𝑥 + 𝑏𝑥 + 𝑎𝑏.
Lesson 7:
Date:
Algebraic Expressions—The Commutative and Associative Properties
4/30/17
S.36
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
36
Lesson #:
Lesson Description
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
Lesson 7A: Operations on Algebraic Expressions
Recall the following rules of exponents:
Product of
Powers
For any number a, and all integers m and n,
am  an = am+n
Power of
a Power
For any number a, and all integers m and n,
(am)n = amn
Power of
a Product
For all numbers a and b, and any integer m,
(ab)m = ambm
Power of
a Monomial
For all numbers a and b, and all integers m, n, and p,
(ambn)p = ampbnp
Quotient of
Powers
For all integers m and n, and any nonzero number a,
am ÷ an = am - n
Zero
Exponent
For any nonzero number a,
a0 = 1
Negative
Exponents
For any nonzero number a and any integer n,
a-n = 1
an
Study the examples below.
(83)5 = (83)(83)(83)(83)(83)
= 83 + 3 + 3+ 3 + 3
= 815
85 = (8 )(8)(8)(8)(8)
83
(8)(8)(8)
= 82
product of powers
y3 = _yyy_ =
y7 yyyyyyy
Lesson 7:
Date:
(y7)3 = (y7)(y7)(y7)
= y7 + 7 + 7
= y21
1 or y-4
y4
Algebraic Expressions—The Commutative and Associative Properties
4/30/17
S.37
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
37
Lesson #:
Lesson Description
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
A Monomial is in Simplest Form when:
 There are no powers of powers
 Each base appears once
 All fractions are in simplest form
 No negative exponents
Simplify each expression
a2a3 =
a2a4 =
a2a5 =
a4a7a =
(3c2)(2c3) =
5457 =
(xy4)(x2x3) =
(8y6)(9y7) =
(-5a4)(6a) =
(2x4)(4x3y2)(-3xy3) =
(63)5 =
(-6  5)2 =
(a4)2(b7)2(c)3 =
(z2y3)2 =
(4xy)3 =
Dividing Monomials
y4 ÷ y 1
(3x4)2  x5 =
z3 ÷ z2
105 ÷ 103
25a5 ÷ 5a2
Lesson 7:
Date:
(-2a2bc)3 =
(-9)(-9z5)2 =
x3 ÷ x3
x3 ÷ x4
Algebraic Expressions—The Commutative and Associative Properties
4/30/17
S.38
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
38
Lesson #:
Lesson Description
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
Problem Set
Write the expression as a single power of the base.
1. 42  43
2. (-3)2(-3)
3. x  x2
4. a2a3a4
5. (42)3
6. [(-3)5]2
7. (a2)5
8. (x3)3
3. (2w)6
4. (4y2z)3
Simplify the expressions.
1. (-3  2)2
2. (2  4)3
Write the expression as a single power of the base.
1. a2a3 =
2. (c2c3)3 =
3. (x)(x3) =
4. 5457 =
5. (y6)(y7) =
6. (-5)4(-5)2 =
7. x4x3x =
8. (a2)2 =
Complete the statement.
1. w7w? = w10
2. (7?)5 = 715
3. [(-2)3]2 = -2?
Transform each given expression into an equivalent one with a positive exponent
1. 4-3
5. 1
2-5
2. 6(3-3)
6. 3-6 ÷ 3-2
Lesson 7:
Date:
3. 10-1
4. 27 ÷ 2-3
7. (5)-2
8. (x4)-3
Algebraic Expressions—The Commutative and Associative Properties
4/30/17
S.39
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
39
Lesson #:
Lesson Description
Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA I
9. 24y4
6y
10.
-65z3
13z2
13. 21a2b
3ab
14. 3ab
3ab
11. -21a5b4
-3a4b
12. 12y2z2
-4y2z
15. 24x3y4
-6xy
16. - 6xy
6xy
17. Simplify
a.
(16𝑥 2 ) ÷ (16𝑥 5 )
b. (2𝑥)4 (2𝑥)3
c.
(9𝑧 −2 )(3𝑧 −1 )−3
d. ((25𝑤 4 ) ÷ (5𝑤 3 )) ÷ (5𝑤 −7 )
e.
(25𝑤 4 ) ÷ ((5𝑤 3 ) ÷ (5𝑤 −7 ))
Lesson 7:
Date:
Algebraic Expressions—The Commutative and Associative Properties
4/30/17
S.40
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
40
Lesson #:
Lesson Description